The non-interacting Hamiltonian (REF ) obeys chiral symmetry [1]}: matrix elements only connect \(A\)  and \(B\)  sites (not \(A\)  to \(A\)  or \(B\)  to \(B\) ) and chiral symmetry ensures particle-hole symmetry of the electronic spectrum. When the spectrum is gapped due to interactions, which generally break chiral symmetry, we use the wave functions in \({\bf k}\)  space to determine the non-Abelian Berry curvature [2]} for the occupied valence bands.
